Conditional Moment Closure Method Research Articles

LES-CMC of a dilute acetone spray flame

A combined Large-Eddy-Simulation (LES) – Conditional Moment Closure (CMC) method has been applied to model spray combustion. Additional source terms in the CMC transport equation that originate from the evaporation of the fuel have been modelled and their influence on flame structure and global quantities such as temporally averaged temperature profiles has been analysed. The predictions of a laboratory spray jet flame are generally good, the temperature profiles are in good agreement with measurements, and predictions of droplet velocities are satisfactory. The evaporation source term leads to a flux in mixture fraction space and peak temperatures are shifted towards higher mixture fraction regions. The results, however, also highlight some limitations associated with mixture fraction as a single conditioning variable for mixture fraction based approaches if applied to liquid fuel combustion.

Parametric study of the Incompletely Stirred Reactor modeling

The Incompletely Stirred Reactor (ISR) is a generalization of the widely-used Perfectly Stirred Reactor (PSR) model and allows for incomplete mixing within the reactor. Its formulation is based on the Conditional Moment Closure (CMC) method. This model is applicable to nonpremixed combustion with strong recirculation such as in a gas turbine combustor primary zone. The model uses the simplifying assumptions that the conditionally-averaged reactive-scalar concentrations are independent of position in the reactor: this results in ordinary differential equations in mixture fraction space. The simplicity of the model permits the use of very complex chemical mechanisms. The effects of the detailed chemistry can be found while still including the effects of micromixing. A parametric study is performed here on an ISR for combustion of methane at overall stoichiometric conditions to investigate the sensitivity of the model to different parameters. The focus here is on emissions of nitric oxide and carbon monoxide. It is shown that the most important parameters in the ISR model are reactor residence time, the chemical mechanism and the core-averaged Probability Density Function (PDF). Using several different shapes for the core-averaged PDF, it is shown that use of a bimodal PDF with a low minimum at stoichiometric mixture fraction and a large variance leads to lower nitric oxide formation. The ‘rich-plus-lean’ mixing or staged combustion strategy for combustion is thus supported.

Modeling of scalar mixing in turbulent jet flames by multiple mapping conditioning

Multiple mapping conditioning (MMC) combines the probability density function (PDF) and the conditional moment closure (CMC) methods via the application of a generalized mapping function to a prescribed reference space. Stochastic and deterministic formulations of MMC exist, and the deterministic implementation has been applied here to a piloted jet diffusion flame (Sandia Flame D). This paper focuses on the feasibility of MMC and its closures for real (laboratory) flames and a relatively simple one-dimensional reference space that represents mixture fraction has been used. The remaining chemically reactive species are implicitly conditioned on mixture fraction and their fluctuations around the conditional mean are neglected. This work primarily evaluates the ability of the deterministic form of MMC to provide accurate and consistent closures for the mixture fraction PDF and the conditional scalar dissipation which do not rely on presumed shape functions for the PDF such as the commonly used β-PDF. Computed probability distributions agree well with measurements, and a detailed comparison of the modeled conditional and mean scalar dissipation with experimental data and conventional closures demonstrate MMC’s potential. Predictions of reactive species and temperature are in good agreement with experimental data and similar in quality to singly conditioned, first-order CMC predictions. MMC therefore provides an attractive-since consistent-alternative approach for the modeling of scalar mixing in turbulent reacting flows.

Second-Order Conditional Moment Closure Simulations of Autoignition of an n-heptane Plume in a Turbulent Coflow of Heated Air

Autoignition of an n-heptane plume in a turbulent coflow of heated air has been studied using the conditional moment closure (CMC) method with a second-order closure for the conditional chemical source term. Two different methodologies have been considered: (i) the Taylor expansion method, in which the second order correction was based on the solution of the full covariance matrix for the 31 reactive species in the chemical mechanism and hence was not limited to a few selected reactions, and (ii) the conditional PDF method, in which only the temperature conditional variance equation has been solved and its PDF assumed to be a β-function. The results compare favorably with experiment in terms of autoignition location. The structure of the reaction zone in mixture fraction space has been explored. The relative performance of the two methodologies is discussed.

First- and second-order elliptic conditional moment closure calculations of piloted methane diffusion flames

Presented are results obtained from the application of a first- and higher-order conditional moment closure (CMC) approach to the modeling of three methane diffusion flames at differing levels of local extinction. In addition to the analysis of higher-order chemistry applications, the results obtained from an elliptic formulation of the CMC equation are considered next to parabolic results presented in previous work. All predictions are based upon second-moment turbulence and scalar-flux closures, and the chemistry applied to represent mean production rates of species is a 16-step reduced mechanism. A second-order chemistry is implemented on the basis of a two-step kinetic representation of methane combustion, used to correct first-order rates. In general, predictions obtained using the second-order model improve significantly upon first-order results for both major and minor species under fuel-rich conditions. The simplified chemistry employed does not however fully capture the effects of local extinction, and suggestions are made regarding the further developments required to permit the accurate prediction of highly strained flames using CMC methods.

Numerical investigation for combustion characteristics of vacuum residue (VR) in a test furnace

It has become inevitable to search for alternative fuels due to current worldwide energy crisis. In this paper combustion characteristics of vacuum residue (VR) is investigated numerically against experimental data in typical operating conditions of a furnace. Heat release reaction is modeled as sequential steps of devolatilization, simplified gas phase reaction and char oxidation as for pulverized coal. Thermal and fuel NO are predicted by the conditional moment closure (CMC) method for estimation of elementary reaction rates. It turns out that Sauter mean diameter (SMD) of VR droplets is a crucial parameter for better combustion efficiency and lower NO. Reasonable agreement is achieved for spatial distributions of major species, temperature and NO for all test cases with different fuel and steam flow rates.

LES-CMC simulations of a turbulent bluff-body flame

The large Eddy simulations (LES)-conditional moment closure (CMC) method with detailed chemistry is applied to a bluff-body stabilized flame. Computations of the velocity and mixture fraction fields show good agreement with the experiments. Temperature and major species are well-predicted throughout the flame with the exception of the flow regions in the outer shear layer close to the nozzle where the pure mixing between hot recirculating products and fresh oxidizer cannot be captured. LES-CMC generally improves on results obtained with RANS-CMC and on LES that uses one representative flamelet to model the dependence of reactive species on mixture fraction. Simulated CO mass fractions are generally in good agreement with the experimental data although a 10% overprediction can be found at downstream positions. NO predictions show a distinct improvement over the flamelet approach, however, simulations overpredict NO mass fractions at all downstream locations due to an overprediction of temperature close to the nozzle. The potential of LES-CMC to predict unsteady finite rate effects is demonstrated by the prediction of endothermic—or “flame cooling”—regions close to the neck of the recirculation zone that favours ethylene production via the methane fuel decomposition channel.

Modeling extinction and reignition in turbulent flames

The conditional moment closure method (CMC) has been extended to improve reactive species predictions in flames with significant local extinction and reignition. Simple first-order closure of the conditionally averaged reaction rate term does not give satisfactory results due to large fluctuations around the conditional mean and an alternative closure is suggested here. The new closure is based on a precomputed parameterized reference field that maps reactive species mass fractions as functions of mixture fraction and sensible enthalpy. During the computations, the reference field is continuously adjusted to ensure consistency with the CMC solution and doubly conditioned chemical source terms that are functions of time, space, mixture fraction, and sensible enthalpy can thus be obtained. Integration over sensible enthalpy space yields the improved singly conditioned chemical source term that can be used for the solution of the CMC equations. Full closure can be achieved by assuming a β-PDF for the probability distribution in sensible enthalpy space and an additional conditional variance equation needs to be solved. The overall agreement between the measured and the computed variance is satisfactory and the extended CMC model is applied to Sandia Flames D, E, and F. Excellent predictions of temperature, major species, intermediates, and NO are obtained in Flames D and E while temperature predictions can be significantly improved in Sandia Flame F.

A joint SSPDF–CMC method for simulating turbulent methane–air jet combustion

A joint single scalar probability density function and conditional moment closure (SSPDF–CMC) method is proposed for modeling a turbulent methane–air jet flame. In general, the probability density function (PDF) of passive scalar (such as mixture fraction) is non-Gaussian and not fully determined by the advecting velocity field, therefore the presumed shape of PDF of mixture fraction assumed as clipped Gaussian distribution or beta function in normal conditional moment closure (CMC) method is incorrect. In SSPDF–CMC method, the PDF of mixture fraction is obtained using a Monte-Carlo method to solve a PDF transport equation. An assumption that the averaged scalar advection is approximately equal to the averaged scalar dissipation in the wake of a grid-generated turbulence flow is adopted to model the averaged scalar dissipation. The predictions using the proposed method are compared with those using the conventional CMC method and the experimental data. It is seen that the predicted Favre conditional averaged statistics and Favre unconditional averaged statistics using the proposed method are in better agreement with the measurement data than those using the conventional CMC method. The predicted conditional or unconditional mean NO even using the SSPDF model is only in fair agreement with the experiments. It shows that the first-order closure for the conditional reaction rate of NO should be improved.

Double conditioning of reactive scalar transport equations in turbulent nonpremixed flames

Double conditioning of reactive scalar transport equations on mixture fraction and sensible enthalpy is presented as a promising extension to conventional conditional moment closure (CMC) methods. Simple first order, singly conditioned CMC cannot accurately predict combustion phenomena such as partially premixed flames and flames with local extinction and reignition. CMC is applied to turbulent flames with significant temperature fluctuations that bring to light the deficiencies of the conventional CMC approach. Fluctuations of temperature and species around their conditional means cannot be neglected when these quantities are conditioned on mixture fraction only. Comparison with direct numerical simulation (DNS) results demonstrates, however, the great potential for doubly conditioned CMC. Doubly conditioned CMC gives excellent agreement with DNS at all times and it captures phenomena like local extinction and the onset of reignition accurately.

IMPERFECTLY STIRRED REACTOR MODEL PREDICTIONS OF REACTION IN A BURNER WITH STRONG RECIRCULATION

An imperfectly stirred reactor (ISR) is a model that can be applied to strongly recirculating reactive flows. This model, which is based on the conditional moment closure method, uses the simplifying approximation that conditionally averaged reactive scalars are independent of position in the reactor. This approximation results in simple ordinary differential equations in mixture fraction space. The simplicity of the model permits use of quite complex chemical kinetics with a relatively small computational cost. In this work the ISR model is applied to an experimental combustor. The first step in the modeling is to obtain the mixture fraction PDFs at positions inside the reactor. To do this, flow and mixing fields are calculated using a computational fluid dynamics code based on a k–ε model and fast chemistry method. Results are then used as input to the ISR code. Mean values of the reactive scalars are compared with experimental data obtained from measurements performed on a burner at the University of Sydney. The comparison shows good agreement in the case of CO, CO2, and temperature, but the methodology still needs to be improved.

Possible Effects of Small-Scale Intermittency in Turbulent Reacting Flows

It is now well established that quantities such as energy dissipation, scalar dissipation and enstrophy possess huge fluctuations in turbulent flows, and that the fluctuations become increasingly stronger with increasing Reynolds number of the flow. The effects of this small-scale “intermittenc” on various aspects of reacting flows have not been addressed fully. This paper draws brief attention to a few possible effects on reaction rates, flame extinction, flamelet approximation, conditional moment closure methods, and so forth, besides commenting on possible effects on the resolution requirements of direct numerical simulations of turbulence. We also discuss the likelihood that large-amplitude events in a given class of shear flows are characteristic of that class, and that, plausible estimates of such quantities cannot be made, in general, on the hypothesis that large and small scales are independent. Finally, we briefly describe some ideas from multifractals as a potentially useful tool for an economical handling of a few of the problems touched upon here.

The Premixed Conditional Moment Closure Method Applied to Idealized Lean Premixed Gas Turbine Combustors

This paper presents the premixed conditional moment closure (CMC) method as a new tool for modeling turbulent premixed combustion with detailed chemistry. By using conditional averages the CMC method can more accurately model the affects of the turbulent fluctuations of the temperature on the reaction rates. This provides an improved means of solving a major problem with traditional turbulent reacting flow models, namely how to close the reaction rate source term. Combined with a commercial CFD code this model provides insight into the emission formation pathways with reasonable runtimes. Results using the full GRI2.11 methane kinetic mechanism are compared to experimental data for a backward-facing step burning premixed methane. This model holds promise as a design tool for lean premixed gas turbine combustors.

The modeling of turbulent reactive flows based on multiple mapping conditioning

A new modeling approach—multiple mapping conditioning (MMC)—is introduced to treat mixing and reaction in turbulent flows. The model combines the advantages of the probability density function and the conditional moment closure methods and is based on a certain generalization of the mapping closure concept. An equivalent stochastic formulation of the MMC model is given. The validity of the closuring hypothesis of the model is demonstrated by a comparison with direct numerical simulation results for the three-stream mixing problem.

Prediction of carbon monoxide in fires by conditional moment closure

Carbon monoxide is the chief killer in fires. Dangerous levels of CO can occur when reacting combustion gases are quenched by heat transfer, or by mixing of the fire plume in a cooled under- or overventilated upper layer. In this paper, carbon monoxide predictions for enclosure fires are modeled by the conditional moment closure (CMC) method and are compared with laboratory data. The modeled fire situation is a buoyant, turbulent, diffusion flame burning under a hood. The fire plume entrains fresh air, and the postflame gases are cooled considerably under the hood by conduction and radiation, emulating conditions which occur in enclosure fires and lead to the freezing of CO burnout. Predictions of CO in the cooled layer are presented in the context of a complete computational fluid dynamics solution of velocity, temperature, and major species concentrations. A range of underhood equivalence ratios, from rich to lean, are investigated. The CMC method predicts CO in very good agreement with data. In particular, CMC is able to correctly predict CO concentrations in lean cooled gases, showing its capability in conditions where reaction rates change considerably.

Modelling Differential Diffusion in Nonpremixed Reacting Turbulent Flow: Application to Turbulent Jet Flames

The modelling of differential diffusion effects in a well documented turbulent hydrogen-air jet diffusion flame is considered- The species and enthalpy transport equations are modelled by the conditional moment closure (CMC) method. In the presence of differential diffusion the terms involving the fluctuations around the conditional means, the eyterms, must not be neglected and need modelling. They have been modelled by an expression which is described in detail in a companion paper. Using these non-unity Lewis number modelling strategies, super-equilibrium temperatures on the lean side of stoichiomelric and increased nitric oxide concentrations are predicted giving improved agreement with experiment. The results show that the eymodel which has been derived with the aid of DNS is applicable to real flame geometries. However, it is also demonstrated that differential diffusion is not the only cause of substantial underpredictions of WOclose to the jet exit. This is in contrast to all previous studies on these hydrogen flames.

Modelling Differential Diffusion in Nonpremixed Reacting Turbulent Flow: Model Development

The modelling of differential diffusion in nonpremixed reacting turbulent flows is considered. The present approach is based on the conditional moment closure (CMC) method. As in the nonreacting case the terms involving the deviations from the conditional averages, the eyterms, cannot be neglected and need modelling. The development of new closures for the eyterms and their validation is aided by DNS studies. Non-unity Schmidt number effects of reacting scalars in homogeneous iso-tropic decaying turbulence are investigated. Chemical kinetics are approximated by irreversible one-step temperature independent and temperature dependent reactions. It is found that the ey-terms and their modelling in reacting flows strongly resemble ey-terms and -models in nonreacting flows (Kronenburg and Bilger, 1997). However, a new definition of a suitable differential diffusion variable is necessary for accurate enclosure in reacting flows. The dependence of the ey-modelling on reaction rate and temperature effects is analysed. CMC generally leads to good predictions of the appropriate conditional average differential diffusion variable which can characterize differential diffusion effects on either species concentrations or enthalpy.

Modeling soot formation in turbulent methane–air jet diffusion flames

The modeling of soot formation and oxidation by the conditional moment closure (CMC) method is considered. It is particularly focused on the influence of differential diffusion of the soot particles on soot predictions. Most importantly, no changes are made to the soot models that were derived from laminar flame experiments and calculations. Good to excellent predictions are achieved in lightly sooting turbulent methane–air jet diffusion flames at atmospheric and elevated pressure when differential diffusion is taken into account. Unity Lewis number assumptions yield underpredictions of soot volume fractions by about 40%. Soot oxidation by OH and O 2 can be treated accurately and both oxidation mechanisms are found to be important for soot burnout in downstream regions. © 2000 by The Combustion Institute

Computation of Conditional Average Scalar Dissipation in Turbulent Jet Diffusion Flames

The modelling of conditional scalar dissipation in locally self-similar turbulent reacting jets is considered. The streamwise dependence in the transport equation of the conserved scalar pdf is represented by a function solely dependent on centreline mixture fraction. This procedure provides a simple model suitable for non-homogeneous flows and ensures positive values for conditional scalar dissipation. It has been tested in pure hydrogen-air jet diffusion flames using a Conditional Moment Closure method with detailed 12species, 23 reactions chemistry. The calculations show good agreement of the averaged scalar dissipation with reference values and the model proves to be superior to previous models based on homogeneous flows if the distribution of the conditional scalar dissipation in mixture fraction space is compared with experimental results. A dependence of NO predictions on the model of conditional scalar dissipation can be observed.

Nitric oxide formation in dilute hydrogen jet flames: isolation of the effects of radiation and turbulence-chemistry submodels

Nitric oxide (NO) levels in turbulent flames are sensitive to temperature, O atom concentrations, and residence times at the NO forming conditions. Consequently, accurate prediction of NO formation is a challenge for turbulent combustion models, and NO is a useful scalar to consider when evaluating turbulence-chemistry submodels. Predictions of NO formation in a series of hydrogen flames with varying levels of helium dilution are compared with experimental results. Dilution reduces flame radiation to the point where uncertainties in the radiation calculation have only a small influence on NO emission, and this allows different models for the coupling of turbulence and chemistry to be compared in relative isolation from the effects of radiation. Calculations using the Probability Density Function (PDF) method and the Conditional Moment Closure (CMC) method are carried out using the same turbulence model, radiation model, reduced chemical mechanism, and boundary conditions, so that similarities and differences between these two approaches for the coupling of turbulence and chemistry may be isolated and more fully understood. To accomplish this, flow model constants are adjusted to match measured axial profiles of mean velocity and mixture fraction in the undiluted flame, before detailed scalar comparisons are carried out. The optically thin radiation assumption is shown to be appropriate for these hydrogen flames, and the effects of flame radiation on NO emission are examined. The sensitivities of the calculated results to certain boundary conditions and submodels are also quantified. Both turbulence-chemistry models yield good results for conditional mean temperature and H 2O mole fraction. Results also show that both models yield quantitatively useful predictions for the NO formation and emission. With all other submodels being the same, the PDF method predicts lower NO levels than does the CMC method. This difference is associated with lower O atom concentrations predicted by the PDF method. Both methods underpredict NO formation in the lower portion of the jet flames. This deficiency is attributed to the effects of differential diffusion, which are not included in the models, and to the influence of heat release on turbulence structure near the flame base.